Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T00:45:26.088Z Has data issue: false hasContentIssue false
  • English
  • Français

On the distribution of random lines

Published online by Cambridge University Press:  14 July 2016

Peter Hall
Peter Hall*
Affiliation:
The Australian National University
*
∗Postal address: Department of Statistics, SGS, The Australian National University, P.O. Box 4, Canberra, A.C.T. 2600, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Draw two lines, L1 and L2, in a plane. Along L1 place the points of a Poisson process, and through each point draw a line, the angles of intersection with L1 being distributed independently and uniformly on (0, π) . The intercepts of these random lines with L2 form a new process, which is Poisson if and only if L1 and L2 are parallel. Curiously, if L1 and L2 are inclined then, with probability 1, the new process forms a dense subset of L2. It is not really even a point process. In the present paper we shall investigate this anomaly and some of its generalizations.

Keywords

LIMIT THEOREMPOISSON POINT PROCESSRANDOM ANGLERANDOM LINE
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 3 , September 1981 , pp. 606 - 616
Copyright
Copyright © Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
Miles, R. E. (1964a) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. U.S.A. 52, 901–907.Google Scholar
Miles, R. E. (1964b) Random polygons determined by random lines in a plane, II. Proc. Nat. Acad. Sci. U.S.A. 52, 1157–1160.CrossRefGoogle Scholar
Moran, P. A. P. (1969) A second note on recent research in geometrical probability. Adv. Appl. Prob. 1, 73–89.Google Scholar
Morton, R. R. A. (1966) The expected number and angle of intersections between random curves in a plane. J. Appl. Prob. 3, 559–562.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar